TIFN-CODAS - Üçgensel Sezgisel Bulanık Sayı CODAS (Daami Remadi & Frikha 2023)

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

1. Steps 1-2 (paper Part i) — Build the linguistic decision matrix X^(l) for each DM d_l (l=1..y) over n alternatives × m criteria; transform linguistic ratings + linguistic criterion weights + DM weights to Triangular Intuitionistic Fuzzy Numbers (TIFN) via fixed 7-point scale (paper Tables 1-2). TIFN representation: x̂_ij^(l) = {(x_ij^1, x_ij^2, x_ij^3); (x'_ij^1, x_ij^2, x'_ij^3)}.

   $$Linguistic-to-TIFNmap(DaamiRemadi\&Frikha2023Table1,7-point):VL=(0,0,0.5);(0,0,0.5)or(0.5,0.5,0.5);(0.5,0.5,0.5)(papervariant—anchoratscaleboundary)L=(0,1,3);(0,1,4)ML=(1,3,5);(0.5,3,5.5)M=(3,5,7);(2,5,8)MH=(5,7,9);(4.5,7,9.5)H=(7,9,10);(6,9,10)VH=(9,10,10);(8,10,10)TIFNorderinginvariant:a^{\prime }1\le a1\le a2\le a3\le a^{\prime }3.$$
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   Linguistic-to-TIFN map (Daami Remadi & Frikha 2023 Table 1, 7-point): VL = {(0,0,0.5); (0,0,0.5)} or {(0.5,0.5,0.5); (0.5,0.5,0.5)} (paper variant — anchor at scale boundary) L = {(0,1,3); (0,1,4)} ML = {(1,3,5); (0.5,3,5.5)} M = {(3,5,7); (2,5,8)} MH = {(5,7,9); (4.5,7,9.5)} H = {(7,9,10); (6,9,10)} VH = {(9,10,10); (8,10,10)} TIFN ordering invariant: a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3.

2. Step 3 (paper Part ii) — For each DM, compute the TIFN normalized decision matrix n̂_ij. Benefit criteria divide by x_j+ = max_i x_ij^'3 (Eq. 13); cost criteria use complementary form with x_j- = min_i x_ij^'1 (Eq. 14). Normalization is component-wise on all 6 TIFN parameters.

   $$Benefit(j\in Nb):n{̂}_{i}j=(x_i{j}^1/x_j+,x_i{j}^2/x_j+,x_i{j}^3/x_j+);(x_i^{\prime }{j}^1/x_j+,x_i{j}^2/x_j+,x_i^{\prime }{j}^3/x_j+)[Eq.(13)]where{x}_j+=ma{x}_i{x}_i{j}^{{}^{\prime }}3Cost(j\in Nc):n{̂}_{i}j=(x_j-/x_i{j}^3,x_j-/x_i{j}^2,x_j-/x_i{j}^1);(x_j-/x_i^{\prime }{j}^3,x_j-/x_i{j}^2,x_j-/x_i^{\prime }{j}^1)[Eq.(14)]where{x}_j-=mi{n}_i{x}_i{j}^{{}^{\prime }}1$$
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   Benefit (j ∈ Nb): n̂_ij = {(x_ij^1/x_j+, x_ij^2/x_j+, x_ij^3/x_j+); (x'_ij^1/x_j+, x_ij^2/x_j+, x'_ij^3/x_j+)} [Eq.(13)] where x_j+ = max_i x_ij^'3 Cost (j ∈ Nc): n̂_ij = {(x_j-/x_ij^3, x_j-/x_ij^2, x_j-/x_ij^1); (x_j-/x'_ij^3, x_j-/x_ij^2, x_j-/x'_ij^1)} [Eq.(14)] where x_j- = min_i x_ij^'1

3. Step 4 — Compute the weighted normalized matrix r̂_ij = w_j ⊗ n̂_ij via TIFN scalar multiplication (Eq. 7-8). When w_j is itself a TIFN (linguistic weight, paper canon), use full TIFN multiplication; when w_j is crisp, the scalar form (Eq. 8) applies component-wise.

   $$ScalarTIFNmultiplication(k>0):k*A_{T}IFN=(k*a1,k*a2,k*a3);(k*a^{\prime }1,k*a2,k*a^{\prime }3)[Eq.(8)]TIFN-TIFNmultiplication(A*B):A*B=(a1*b1,a2*b2,a3*b3);(a^{\prime }1*b^{\prime }1,a2*b2,a^{\prime }3*b^{\prime }3)[Eq.(7)]Weightedmatrix:r{̂}_{i}j=w_j\otimes n{̂}_{i}j\forall i=1..n,j=1..m[Eq.(15)]$$
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   Scalar TIFN multiplication (k > 0): k * A_TIFN = {(k*a1, k*a2, k*a3); (k*a'1, k*a2, k*a'3)} [Eq.(8)] TIFN-TIFN multiplication (A * B): A * B = {(a1*b1, a2*b2, a3*b3); (a'1*b'1, a2*b2, a'3*b'3)} [Eq.(7)] Weighted matrix: r̂_ij = w_j ⊗ n̂_ij ∀ i=1..n, j=1..m [Eq.(15)]

4. Step 5 — Determine the TIFN negative-ideal solution tns_j for each criterion j. Selection rule: tns_j = r̂_(i*)j where i* = argmin_i r'_ij^1 (smallest left-of-ν boundary). Vector tn̂s = [tns_1, ..., tns_m].

   $$tn̂s=[tn̂s_j{]}_{1\times m}[Eq.(16)]tn̂s_j=(n{s}_j^1,n{s}_j^2,n{s}_j^3);(n{s_j^{\prime }}^1,n{s}_j^2,n{s_j^{\prime }}^3)=mi{n}_{i}r{̂}_{i}jwheremi{n}_{i}r{̂}_{i}jselectsther{̂}_{i}jwiththelowest{r}_i^{\prime }{j}^1.$$
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   tn̂s = [tn̂s_j]_{1×m} [Eq.(16)] tn̂s_j = {(ns_j^1, ns_j^2, ns_j^3); (ns'_j^1, ns_j^2, ns'_j^3)} = min_i r̂_ij where min_i r̂_ij selects the r̂_ij with the lowest r'_ij^1.

5. Step 6 — Defuzzify the TIFN weighted normalized matrix and the TIFN negative-ideal solution to crisp scalars via the Gani-Abbas (2014) weighted-average defuzzifier (Eq. 10). Each TIFN element A = {(a1,a2,a3);(a'1,a2,a'3)} maps to A_d ∈ ℝ.

   $$Defuzzifier(Gani-Abbas2014):A_d=((a1+2a2+a3)+(a^{\prime }1+2a2+a^{\prime }3))/8[Eq.(10)]Weightedmatrixdefuzzification:r_i^{d}j=((r_i{j}^1+2r_i{j}^2+r_i{j}^3)+(r_i^{\prime }{j}^1+2r_i{j}^2+r_i^{\prime }{j}^3))/8[Eq.(17)]NISdefuzzification:n{s}_j^d=((n{s}_j^1+2n{s}_j^2+n{s}_j^3)+(n{s_j^{\prime }}^1+2n{s}_j^2+n{s_j^{\prime }}^3))/8[Eq.(18)]$$
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   Defuzzifier (Gani-Abbas 2014): A_d = ((a1 + 2a2 + a3) + (a'1 + 2a2 + a'3)) / 8 [Eq.(10)] Weighted matrix defuzzification: r^d_ij = ((r_ij^1 + 2 r_ij^2 + r_ij^3) + (r'_ij^1 + 2 r_ij^2 + r'_ij^3)) / 8 [Eq.(17)] NIS defuzzification: ns^d_j = ((ns_j^1 + 2 ns_j^2 + ns_j^3) + (ns'_j^1 + 2 ns_j^2 + ns'_j^3)) / 8 [Eq.(18)]

6. Step 7 — Apply classical CODAS (Keshavarz Ghorabaee 2016) on the defuzzified matrix r^d_ij and NIS ns^d_j. Compute the Euclidean distance E_i (Eq. 1, L2 norm) and Taxicab distance T_i (Eq. 2, L1 norm) of each alternative from the negative-ideal. Build the n×n relative assessment matrix R_a using threshold function ψ (default τ = 0.02) per Eq. 19; sum row-wise to obtain individual assessment score H_i (Eq. 20).

   $$Euclideandistance:E_i=sqrt({\Sigma }_{j=1}^m(r_i^{d}j-n{s}_j^d{)}^2)[Eq.(1)]Taxicabdistance:T_i={\Sigma }_{j=1}^m|r_i^{d}j-n{s}_j^d|[Eq.(2)]Relativeassessmentmatrix:h_{i}k=(E_i-E_k)+\psi (E_i-E_k)\times (T_i-T_k)[Eq.(19)]\psi (x)=1if|x|\ge \tau ;0otherwise(\tau default=0.02perKeshavarzGhorabaee2016)Assessmentscore:H_i={\Sigma }_{k=1}^n{h}_{i}k[Eq.(20)]$$
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   Euclidean distance: E_i = sqrt( Σ_{j=1}^m (r^d_ij − ns^d_j)^2 ) [Eq.(1)] Taxicab distance: T_i = Σ_{j=1}^m | r^d_ij − ns^d_j | [Eq.(2)] Relative assessment matrix: h_ik = (E_i − E_k) + ψ(E_i − E_k) × (T_i − T_k) [Eq.(19)] ψ(x) = 1 if |x| ≥ τ ; 0 otherwise (τ default = 0.02 per Keshavarz Ghorabaee 2016) Assessment score: H_i = Σ_{k=1}^n h_ik [Eq.(20)]

7. Step 8 (paper Part iii) — Aggregate individual assessment scores across l decision makers using DM weights λ_l (Σ_l λ_l = 1) into the group assessment score HG_i. In single-DM mode (y=1), HG_i = H_i trivially.

   $$H{G}_i={\Sigma }_{l=1}^y{\lambda }_l\times H_i^{(}l)[Eq.(21)]Where{H}_i^{(}l)istheindividualassessmentscoreofalternativeiunderdecision-maker{d}_l,and{\lambda }_{l}isthecrispDMweight.$$
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   HG_i = Σ_{l=1}^y λ_l × H_i^(l) [Eq.(21)] Where H_i^(l) is the individual assessment score of alternative i under decision-maker d_l, and λ_l is the crisp DM weight.

8. Step 9 — Rank alternatives in descending order of group assessment score HG_i. Higher HG_i indicates greater desirability (the alternative is farther from the negative-ideal in the L2/L1 combined metric).

   $$Ranking:sortalternativesbyH{G}_{i}descending.A_{(}{i}_1)\succ A_{(}{i}_2)\succ ...\succ A_{(}{i}_n)\iff H{G}_{i_1}>H{G}_{i_2}>...>H{G}_{i_n}$$
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   Ranking: sort alternatives by HG_i descending. A_(i_1) ≻ A_(i_2) ≻ ... ≻ A_(i_n) ⇔ HG_{i_1} > HG_{i_2} > ... > HG_{i_n}

Sezgi

Combinative Distance-based Assessment under Triangular Intuitionistic Fuzzy uncertainty (TIFN: {(a1,a2,a3); (a'1,a2,a'3)}; a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3) - MCGDM with linguistic-to-TIFN translation. Output typically utility (higher value = preferred).

Sonucu okuma: TIFN-CODAS extends classical CODAS (Keshavarz Ghorabaee 2016) to handle Triangular Intuitionistic Fuzzy uncertainty (Li-Nan-Zhang 2010 TIFN over Atanassov 1986 IFS). Linguistic ratings are mapped to TIFNs via a fixed 7-point scale (paper Table 1). TIFN arithmetic (normalize → weight) is performed in fuzzy space, then defuzzified via Gani-Abbas (2014) weighted-average to crisp scalars before applying CODAS distances (Euclidean L2 + Taxicab L1) from the TIFN negative-ideal. Combined assessment uses threshold ψ (default τ = 0.02) to recognize equivalent Euclidean distances. For MCGDM (l > 1), individual H_i scores are aggregated by DM weights λ. Higher HG_i = better alternative.

Varsayımlar

• Decision matrix entries are valid TIFNs (ordering a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3)
• DM weights λ_l sum to 1 in MAGDM group setting
• All decision-makers use the same linguistic-to-TIFN translation table (paper Tables 1-2)
• τ threshold reflects domain-appropriate Euclidean equality tolerance

Ne zaman kullanılmaz

• Classical data sufficient - use base CODAS directly
• Single-valued IFS already provides enough granularity - use IF-CODAS (Ren 2018)
• Interval bounds on μ/ν needed - use IVIF-CODAS (Boltürk-Kahraman 2018)
• Linguistic granularity coarser than 7 points - consider crisp methods

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: Inherited from crisp CODAS (Keshavarz Ghorabaee 2016); empirical comparison with IVIF-VIKOR/TOPSIS in Daami Remadi & Frikha 2023 Table 10 shows close rankings across methods.)
• Assumes: Decision matrix entries are valid TIFNs (ordering a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3)
• Assumes: DM weights λ_l sum to 1 in MAGDM group setting
• Assumes: All decision-makers use the same linguistic-to-TIFN translation table (paper Tables 1-2)
• Assumes: τ threshold reflects domain-appropriate Euclidean equality tolerance

Sık yapılan hatalar

• Değer-uzayı ihlali: tüm TIFN girişlerinin a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3 sıralamasını sağladığından emin olun; μ-taşıyıcı [a1,a3] ν-taşıyıcı [a'1,a'3] içinde olmalı.
• ψ eşik hassasiyeti: alternatiflerin Öklid uzaklıkları birbirine yakınsa τ değeri sıralamayı önemli ölçüde etkiler. Makale varsayılanı τ = 0.02 (Keshavarz Ghorabaee 2016 önerisi); kullanıcı D.parameters.psi_threshold ile geçersiz kılabilir.
• Defuzzifikasyon sırası: SADECE ağırlıklı normalleştirilmiş matris kurulduktan sonra defuzzifiye edin - ham karar matrisini önce defuzzifiye etmeyin (TIFN-ağırlıklı ölçek bilgisi kaybolur).
• Cost kriter normalizasyonu: Eq. 14'ü x_j- = min_i x_ij^'1 ile kullanın - TIFN tümleyenini (μ↔ν değiştir) uygulamayın; makale tümleyeni değil min/maks normalizasyonunu kullanır.